352 XV NORMED ALGEBRAS AND SPECTRAL THEORY U(s) is a continuous operator on H0 - In view of (15.6.5) and (5.5.1), this con- tinuity is equivalent to the following condition on the form g — gxo: (U) For each s e A there exists a real number Ms ^ 0 such that (15.6.9) g(st,st)^Msg(t, i) for all 16 A. Conversely: (15.6.10) Let g be a positive Hilbertform on A, satisfying the condition (U), and suppose that (with the notation of (15.6.8)) the prehilbert space A/n is separable, so that (6.6.2) it can be identified with a dense vector subspace of a Hilbert space H. Then for each seA the endomorphism n(i) H* n(sf) of A/n extends to a continuous endomorphism x \-> V(s) - x 0/H, and s \-> V(s) is a repre- sentation ofAinVL. If n(f) = 7i(O» then n(st) = n(st)') because n is a left ideal. Hence the endomorphism of H0 under consideration is well-defined, and the definition of the scalar product on A/n, together with (15.6.9), shows that this endomor- phism is continuous (5.5.1). Hence the existence of the continuous operator V(s) (5.5.4). Since n((ss')t) = Ti(s(s'i)\ we have V(ss')= V(s)V(s'). Also, by virtue of (15.6.3) (KO*) • 71(01 *(0) = &**, O = &, *0 = *<X, 0 = (V(s) ' n(t') | TU(O) = (n(t) | V(s) ' which shows that V(s*) = (V(s))*. Finally, if A has a unit element e, then evidently V(e) = 1H , and therefore jh-> V(s) is a representation of A in H. It is useful to know when the representation s\-+ V(s) so defined is non- degenerate (15.5.5). This is equivalent to the following condition on g: (N) The elements n(sf)form a total set in the prehilbert space A/n. This condition is trivially satisfied when A has a unit element. Under the conditions of (1 5.6.1 0), there does not in general exist an element x0 e H such that g(s, t) = (V(s) - x0 \ V(f) - ;c0) (cf. Section 15.9, Problem 3). However, such a vector always exists when A has a unit element e : we may take x0 = n(e).n be reconstructed from the algebra structure of A and the form